Algebras and Representation Theory

, Volume 17, Issue 6, pp 1657–1682 | Cite as

Finite Parabolic Conjugation on Varieties of Nilpotent Matrices

  • Magdalena BoosEmail author


We consider the conjugation-action of an arbitrary upper-block parabolic subgroup of GLn(C) on the variety of x-nilpotent complex matrices and translate it to a representation-theoretic context. We obtain a criterion as to whether the action admits a finite number of orbits and specify a system of representatives for the orbits in the finite case of 2-nilpotent matrices. Furthermore, we give a set-theoretic description of their closures and specify the minimal degenerations in detail for the action of the Borel subgroup. We show that in all non-finite cases, the corresponding quiver algebra is of wild representation type.


Parabolic orbits of nilpotent matrices Degenerations Finite classification 

Mathematics Subject Classification (2010)

16G20 16G60 16W22 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Assem, I., Simson, D., Skowroński, A.: Elements of the Representation Theory of Associative Algebras. Vol. 1, London Mathematical Society Student Texts, vol. 65. Cambridge University Press, Cambridge (2006). doi: 10.1017/CBO9780511614309 CrossRefGoogle Scholar
  2. 2.
    Bongartz, K.: Minimal singularities for representations of Dynkin quivers. Comment. Math. Helv. 69(4), 575–611 (1994). doi: 10.1007/BF02564505 CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Bongartz, K., Gabriel, P.: Covering spaces in representation-theory. Invent. Math. 65(3), 331–378 (1981/82)Google Scholar
  4. 4.
    Boos, M.: Conjugation on varieties of nilpotent matrices. PhD thesis, Bergische Universität, Wuppertal (2012).
  5. 5.
    Boos, M., Reineke, M.: B-orbits of 2-nilpotent matrices. In: Highlights in Lie Algebraic Methods (Progress in Mathematics). Birkhäuser, Boston, pp. 147–166 (2011)Google Scholar
  6. 6.
    Brion, M.: Spherical varieties. In: Proceedings of the International Congress of Mathematicians, vol. 1, 2 (Zürich, 1994). Birkhäuser, Boston, pp. 753–760 (1995)Google Scholar
  7. 7.
    Drozd, Y.A.: Tame and wild matrix problems. In: Representation Theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979), Lecture Notes in Math., vol. 832. Springer, Berlin, pp. 242–258 (1980)Google Scholar
  8. 8.
    Fresse, L.: On the singular locus of certain subvarieties of Springer fibres. To appear in: Mathematical Research Letters (2012)Google Scholar
  9. 9.
    Gabriel, P.: Unzerlegbare Darstellungen. I. Manuscripta Math. 6, 71–103; correction, ibid. 6(1972):309 (1972)Google Scholar
  10. 10.
    Gabriel, P.: The universal cover of a representation-finite algebra. In: Representations of Algebras (Puebla, 1980), Lecture Notes in Math, vol. 903. Springer, Berlin, pp. 68–105 (1981)Google Scholar
  11. 11.
    Gerstenhaber, M.: On nilalgebras and linear varieties of nilpotent matrices. III. Ann. Math. 70(2), 167–205 (1959)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Hesselink, W.: Singularities in the nilpotent scheme of a classical group. Trans. Am. Math. Soc. 222, 1–32 (1976)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Hille, L., Röhrle, G.: A classification of parabolic subgroups of classical groups with a finite number of orbits on the unipotent radical. Transform. Groups 4(1), 35–52 (1999). doi: 10.1007/BF01236661 CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Jordan, M.E. C.: Sur la résulotion des équations différentielles linéaires. Ouevres 4, 313–317 (1871)Google Scholar
  15. 15.
    Jordan, M.E.C.: Traité des substitutions et des équations algébriques. Les Grands Classiques Gauthier-Villars. [Gauthier-Villars Great Classics]. Éditions Jacques Gabay, Sceaux (1989). Reprint of the 1870 originalGoogle Scholar
  16. 16.
    Kraft, H.: Geometrische Methoden in der Invariantentheorie. Aspects of Mathematics, D1. Friedr. Vieweg & Sohn, Braunschweig (1984)Google Scholar
  17. 17.
    Magyar, P., Weyman, J.M., Zelevinsky, A.: Multiple flag varieties of finite type. Adv. Math. 141, 97–118 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Melnikov, A.: B-orbits in solutions to the equation X 2=0 in triangular matrices. J. Algebra 223(1), 101–108 (2000). doi: 10.1006/jabr.1999.8056 CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Melnikov, A.: Description of B-orbit closures of order 2 in upper-triangular matrices. Transform. Groups 11(2), 217–247 (2006). doi: 10.1007/s00031-004-1111-0 CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Panyushev, D.I.: Complexity and nilpotent orbits. Manuscripta Math. 83(3–4), 223–237 (1994). doi: 10.1007/BF02567611 CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Timashev, D.A.: A generalization of the Bruhat decomposition. Izv. Ross. Akad. Nauk Ser. Mat. 58(5), 110–123 (1994). doi: 10.1070/IM1995v045n02ABEH001643 zbMATHGoogle Scholar
  22. 22.
    Vinberg, E.B.: Complexity of actions of reductive groups. Funktsional. Anal. i Prilozhen. 20, 1–13 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Weist, T.: Tree modules. To appear in: Bulletin of the London Mathematical Society. arXiv:1011.1203 (2010)

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Fachbereich C - MathematikBergische Universität WuppertalWuppertalGermany

Personalised recommendations